Question

5.8 Explain how the following conditions can be represented as linear constraints using binary variables. (a) Either x1 + x2 ? 3 or 3x1 + 4x2 ? 10 (b) Variable x2 can assume values 0, 4, 7, 10, and 12 only (c) If x2 ? 3, then x3 ? 6; Otherwise x3 ? 4 (assume x2 and x3 are integers) (d) At least 2 out of the following 5 constraints must be satisfied:

x1 + x2 <=7

x1 ? x2 >=3

2x1 + 3x2 <=20

4x1 ? 3x2 >=10

x2 ? 6

x1, x2 ? 0

Answer #1

a. Solve the following linear programming model by using the
graphical method: graph the constraints and identify the feasible
region then determine the optimal solution (s) (show your
work).
Minimize Z = 3x1 + 7x2
Subject to 9x1 + 3x2 ≥ 36
4x1 + 5x2 ≥ 40
x1 – x2 ≤ 0
2x1 ≤ 13
x1, x2 ≥ 0
b. Are any constraints binding? If so, which one (s)?

Duality Theory: What are all the correct objective function and
constraints? Choose all that apply and consider the following LP:
max 4x1+x2
x1+2x2=6
x1−x2≥3
2x1+x2≤10
x1,x2≥0
Now formulate a dual of this linear program. Select all of the
coefficients that appear in the objective function of the dual
1. 6
2. 2
3. 3
4. 10
5. 1
6. -1
7. 4

Solve the following linear programming model by using the
graphical method: graph the constraints and identify the feasible
region. Using the corner points method, determine the optimal
solution (s) (show your work).
Maximize Z = 6.5x1 + 10x2
Subject to x1 + x2 ≤ 15
2x1 + 4x2 ≤ 40
x1 ≥ 8
x1, x2 ≥ 0
b. If the constraint x1 ≥ 8 is changed to x1 ≤ 8, what effect
does this have on the optimal solution? Are...

Consider a capital budgeting problem with seven projects
represented by binary (0 or 1) variables X1,
X2, X3, X4, X5,
X6, X7.
Write a constraint modeling the situation in which only
2 of the projects from 1, 2, 3, and 4 must be
selected.
Write a constraint modeling the situation in which at
least 2 of the project from 1, 3, 4, and 7 must be
selected.
Write a constraint modeling the situation project 3 or
6 must be selected,...

3. Consider the system of linear equations
3x1 + x2 + 4x3 − x4
= 7
2x1 − 2x2 − x3 + 2x4
= 1
5x1 + 7x2 + 14x3 −
8x4 = 20
x1 + 3x2 + 2x3 + 4x4
= −4
b) Solve this linear system applying Gaussian forward
elimination with partial pivoting and back ward substitution, by
hand. In (b) use fractions throughout your calculations.
(i think x1 = 1, x2= -1, x3 =1,
x4=-1, but i...

Consider the following linear programming
problem.
Maximize 6X1
+ 4X2
Subject to:
X1
+ 2X2 ≤ 16
3X1
+ 2X2 ≤ 24
X1 ≥
2
X1,
X2 ≥ 0
Use Excel Solver to find the optimal values of X1 and
X2. In other words, your decision variables:
a.
(10, 0)
b.
(12, 2)
c.
(7, 5)
d.
(0, 10)

determine the maximum number of basic feasible solution in the
linear program with the following constraints :
x1+x2<=6
x2<=3
x1,x2>=0
note that you will have to introduce two slack variables to the
above constraints

Duality Theory: Consider the following LP:
max 2x1+2x2+4x3
x1−2x2+2x3≤−1
3x1−2x2+4x3≤−3
x1,x2,x3≤0
Formulate a dual of this linear program. Select all the correct
objective function and constraints
1. min −y1−3y2
2. min −y1−3y2
3. y1+3y2≤2
4. −2y1−2y2≤2
5. 2y1+4y2≤4
6. y1,y2≤0

The following is the mathematical model of a linear programming
problem for profit:
Maximize subject to
Z = 2X1 + 3X2
4X1+9X2 ≤ 72 10X1 + 11X2 ≤ 110 17X1 + 9X2 ≤ 153
X1 , X2 ≥ 0
The constraint lines have been graphed below along with one
example profit line (dashed). The decision variable X1 is used as
the X axis of the graph. Use this information for questions
19 through 23.
A). Which of the following gives...

Consider the following linear program Max 5x1+5x2+3x3
St
x1+3x2+x3<=3
-x1+ 3x3<=2
2x1-x2 +2x3<=4
2x1+3x2-x3<=2
xi>=0 for i=1,2,3
Suppose that while solving this problem with Simplex method, you
arrive at the following table:
z
x1
x2
x3
x4
x5
x6
x7
rhs
Row0
1
0
-29/6
0
0
0
11/6
2/3
26/3
Row1
0
0
-4/3
1
0
0
1/3
-1/3
2/3
Row2
0
1
5/6
0
0
0
1/6
1/3
4/3
Row3
0
0
7/2
0
1
0
-1/2
0...

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